THE EXISTENCE OF SOLUTIONS FOR A SHEAR THINNING COMPRESSIBLE NON-NEWTONIAN MODELS

. This paper is concerned with the initial boundary value problem for a shear thinning ﬂuid-particle interaction non-Newtonian model with vac- uum. The viscosity term of the ﬂuid and the non-Newtonian gravitational force are fully nonlinear. Under Dirichlet boundary for velocity and the no-ﬂux con- dition for density of particles, the existence and uniqueness of strong solutions is investigated in one dimensional bounded intervals.

There are many kinds of literatures on the study of the existence and behavior of solutions to Navier-Stokes equations (See [1]- [17]). Taking system (1) as an example, Carrillo et al [4] discussed the the global existence and asymptotic behavior of the weak solutions providing a rigorous mathematical theory based on the principle of balance laws, following the framework of Lions [18] and Feireisl et al [11,12]. Motivated by the stability arguments in [5], the authors also investigated the numerical analysis in [6]. Ballew and Trivisa [1] constructed suitable weak solutions and low stratification singular limit for a fluid particle interaction model. In addition, Mellet and Vasseur [20] proved the global existence of weak solutions of equations by using the entropy method on the asymptotic regime corresponding to a strong drag force and strong brownian motion. Zhang et al [31] establish the existence and uniqueness of classical solution to the system (1) .
Despite the important progress, there are few results of non-Newtonian fluidparticle interaction model. As we know, the Navier Stokes equations are generally accepted as a right governing equations for the compressible or incompressible motion of viscous fluids, which is usually described as ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) − div(Γ) + ∇P = ρf, where Γ denotes the viscous stress tensor, which depends on E ij (∇u), and is the rate of strain. If the relation between the stress and rate of strain is linear, namely, Γ = µE ij (∇u), where µ is the viscosity coefficient, then the fluid is called Newtonian. If the relation is not linear, the fluid is called non-Newtonian. The simplest model of the stress-strain relation for such fluids given by the power laws, which states that for 0 < q < 1 (see [3]). In [16], Ladyzhenskaya proposed a special form for Γ on the incompressible model: For µ 0 = 0, if p < 2 it is a pseudo-plastic fluid. In the view of physics, the model captures the shear thinning fluid for the case of 1 < p < 2 (see [19]).
Non-Newtonian fluid flows are frequently encountered in many physical and industrial processes [8,9], such as porous flows of oils and gases [7], biological fluid flows of blood [30], saliva and mucus, penetration grouting of cement mortar and mixing of massive particles and fluids in drug production [13]. The possible appearance of the vacuum is one of the major difficulties when trying to prove the existence and strong regularity results. On the other hand, the constitutive behavior of non-Newtonian fluid flow is usually more complex and highly non-linear, which may bring more difficulties to study such flows.
In recent years, there has been many research in the field of non-Newtonian flows, both theoretically and experimentally (see [14]- [26]). For example, in [14], Guo and Zhu studied the partial regularity of the generalized solutions to an incompressible monopolar non-Newtonian fluids. In [32], the trajectory attractor and global attractor for an autonomous non-Newtonian fluid in dimension two was studied. The existence and uniqueness of solutions for non-Newtonian fluids were established in [29] by applying Ladyzhenskaya's viscous stress tensor model.

A priori estimates for smooth solutions
In this section, we will prove the local existence of strong solutions. From the continuity equation (2) 1 , we can deduce the conservation of mass Because equation (2) 2 possesses always with singularity, we overcome this difficulty by introduce a regularized process, then by taking the limiting process back to the original problem. Namely, we consider the following system with the initial and boundary conditions.
Provided that (ρ, u, η) is a smooth solution of (10)- (15) and ρ 0 ≥ δ, where 0 < δ 1 is a positive number. We denote by We first get the estimate of |u 0xx | L 2 . From (16), we have Then where C is a positive constant, depending only on M 0 .
Next, we introduce an auxiliary function We will derive some useful estimate to each term of Z(t) in terms of some integrals of Z(t), then apply arguments of Gronwall's inequality to prove Z(t) is locally bounded.
2.1. Preliminaries. In order to prove the main Theorem, we first give some useful lemmas for later use.
Proof. According to (18), we have Taking it by the L 2 norm, we have Therefore, by the above inequality, as ε j → 0, For all α > 0, there exists N , as i, j > N , we can deduce that . With the assumption, we can obtain where C is a positive constant, depending only on |ρ 0 | H 1 (Ω) , |g| L 2 (Ω) and |η 0 | H 2 (Ω) . Using the following inequality, where 0 < θ < 1. By the simple calculation, we can get where C depending only on p, then Substituting this into (18), we have By the uniqueness of the weak convergence, we have This completes the proof of Lemma 2.1.
where C is a positive constant, depending only on M 0 .
Proof. We estimates for u and η for later use. It follows from (11) that We note that Taking it by the L 2 norm and using Young's inequality, we have On the other hand, by (12), we have Taking it by L 2 -norm, using Young's inequality, which gives This implies that By (13), taking it by the L 2 norm, we have Multiplying (10) by ρ, integrating over Ω, we deduce that Integrating it by parts, using Sobolev inequality, we obtain Differentiating (10) with respect to x, and multiplying it by ρ x , integrating over Ω, and using Sobolev inequality, we have From (26) and (27) and the Gronwall's inequality, then lemma 2.2 holds. Lemma 2.3.
where C is a positive constant, depending only on M 0 .
Proof. Multiplying (13) by η, integrating the resulting equation over Ω T , using the boundary conditions (4) and Young's inequality, we have Multiplying (13) by η t , integrating (by parts) over Ω T , using the boundary conditions (4) and Young's inequality, we have Differentiating (13) with respect to t, multiplying the resulting equation by η t , integrating (by parts) over Ω T , we get Combining (29)-(31), we obtain the desired estimate of Lemma 2.3.
where C is a positive constant, depending only on M 0 .
Proof. Using (10), we rewritten the (11) as Multiplying (33) by u t , integrating (by parts) over Ω T , we have We deal with each term as follows: By virtue of (10), we have Substituting the above into (34), using Sobolev inequality and Young's inequality, we have To estimate (36), combining (35) we have the following estimates In exactly the same way, we also have which, together with (36) and (37), implies (32) holds.
where C is a positive constant, depending only on M 0 .
Proof. Differentiating equation (11) with respect to t, multiplying the result equation by u t , and integrating it over Ω, we have Note that Let from (24), it follows that Combining (35), (40) can be rewritten into Using Sobolev inequality, Young's inequality, (11), (24) and (25), we obtain In order to estimate I 11 , we need to deal with the estimate of |Ψ xt | L 2 . Differentiating (12) with respect to t, multiplying it by Ψ t and integrating over Ω, we have and Then (43) can be rewritten into Using Young's inequality, combining the above estimates we deduce that Substituting I j (j = 1, 2, . . . , 11) into (42), and integrating over (τ, t) ⊂ (0, T ) on the time variable, we have To obtain the estimate of | √ ρu t (t)| 2 L 2 , we need to estimate lim τ →0 | √ ρu t (τ )| 2 L 2 . Multiplying (33) by u t and integrating over Ω, we get According to the smoothness of (ρ, u, η), we have Then, taking a limit on τ in (45), as τ → 0, we can easily obtain This complete the proof of Lemma 2.5.
With the help of Lemma 2.2 to Lemma 2.5, and the definition of Z(t), we conclude that where C,C are positive constants, depending only on M 0 . This means that there exist a time T 1 > 0 and a constant C, such that ess sup where C is a positive constant, depending only on M 0 .

Proof of the main theorem
In this section, the existence of strong solutions can be established by a standard argument. We construct the approximate solutions by using the iterative scheme, derive uniform bounds and thus obtain solutions of the original problem by passing to the limit. Our proof will be based on the usual iteration argument and some ideas developed in [10]. Precisely, we first define u 0 = 0 and assuming that u k−1 was defined for k ≥ 1, let ρ k , u k , η k be the unique smooth solution to the following system with the initial and boundary conditions (ρ k , u k , η k )| t=0 = (ρ 0 , u 0 , η 0 ), (53) where With the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold where C is a generic constant depending only on M 0 , but independent of k.
In addition, we first find ρ k from the initial problem with smooth function u k−1 , obviously, there is a unique solution ρ k on the above problem and also we could obtain that (., s)| L ∞ ds > 0, for all t ∈ (0, T 1 ).
Next, we will prove the approximate solution (ρ k , u k , η k ) converges to a limit (ρ ε , u ε , η ε ) in a strong sense. To this end, let us definē By a direct calculation, we can verify that the functionsρ k+1 ,ū k+1 ,η k+1 satisfy the system of equations Multiplying (56) byρ k+1 , integrating over Ω and using Young's inequality, we obtain d dt where C ζ is a positive constant, depending on M 0 and ζ for all t < T 1 and k ≥ 1.